# Isomorphic encryption or homomorphic encryption?

Many encryption functions are said to be homomorphic:

http://en.wikipedia.org/wiki/Homomorphic_encryption

As encryption functions are invertible, they can be considered one-to-one and onto on properly defined domains and ranges.

So, my basic question is why we don't use the term "isomorphic encryption" rather than "homomorphic encryption"?

First, consider the definition of group isomorphism:

Given two groups $(G, \otimes)$ and $(H, \odot)$, a group isomorphism from $(G, \otimes)$ to $(H, \odot)$ is a bijective function $f : G \to H$ such that for all $u$ and $v$ in $G$ it holds that $f(u \otimes v) = f(u) \odot f(v)$.

Now, consider a homomorphic encryption such as ElGamal cryptosystem: It takes a message from a cyclic group $G$, and outputs a pair $(c_1, c_2) \in G^2$. That is, $\mathcal{E} \colon G \to G^2$.

Notice that under this definition, $\mathcal{E}$ is not a bijection from the message space $G$ to the ciphertext space $G^2$. However, the decryption of ElGamal is unique.

Homomorphic Encryption does not necessarily mean that the encryption function is a group homomorphism.

Let $E: M \rightarrow C$ be the encryption function where $M$ and $C$ are the plaintext and ciphertext space respectively. Now $E$ is said to be homomorphic if $E(m_1 * m_2)=E(m_1)\cdot E(m_2)$, where $*$ and $\cdot$ are binray operations on $M$ and $C$ respectively. However, $M$ and $C$ are not necessarily possessing an algebraic structure like groups, rings etc. Thus, even though the encryption function $E$ is invertible, the question of isomorphism does not arise.

• A set endowed with an operation possesses an algebraic structure, even if the structure is not a group. We are sure that the set M is closed under *, otherwise, the "multiplication of two valid plaintexts will not result in a valid plaintext. So, in the very least, we have an algebraic structure called Magma: en.wikipedia.org/wiki/Magma_(algebra). Jun 26 '15 at 2:14